10th Maths Unit 6 Important 5 Mark Questions

Conquer the most challenging chapter of your syllabus by practicing the 10th Maths Unit 6 Important 5 Marks questions. Trigonometry is a highly rewarding, concept-driven unit that regularly features long-answer problems based on proving trigonometric identities and solving real-world applications of heights and distances (using angles of elevation and depression). This expert-curated list highlights the most frequently repeated 5-mark questions and tricky compulsory problems, providing you with the exact formulas and structured steps needed to secure a perfect score in your board exams.

UNIT - 6 (TRIGONOMETRY)

IMPORTANT SUMS (5 MARKS)

1. Prove that sin²A cos²B + cos²A sin²B + cos²A cos²B + sin²A sin²B = 1.

2. If cosec θ + cot θ = P, then prove that cos θ = ^{(P² - 1)}/_{(P² + 1)}.

3. Prove that ^{(1 + cot A + tan A)(sin A - cos A)}/_{(sec³A - cosec³A)} = sin²A cos²A.

4. If √3 sin θ - cos θ = 0, then show that tan 3θ = ^{(3 tan θ - tan³θ)}/_{(1 - 3 tan²θ)}.

5. If ^{cos θ}/_{(1 + sin θ)} = ¹/_a, prove that ^{(a² - 1)}/_{(a² + 1)} = sin θ.

6. If cot θ + tan θ = x and sec θ - cos θ = y then prove that (x²y)^2/3 - (xy²)^2/3 = 1.

7. Two ships are sailing in the sea on either sides of a light house. The angle of elevation of the top of the light house as observed from the ships are 30° and 45° respectively. If the lighthouse is 200 m high, find the distance between the two ships. (√3 = 1.732)

8. From a point on the ground, the angle of elevation of the bottom and top of a tower fixed at the top of a 30 m high building are 45° and 60° respectively. Find the height of the tower. (√3 = 1.732)

9. To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respectively. If the height of the man is 180 cm and if he is 5 m away from the wall, what is the height of the window?

10. A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 40°. Find the height of the pedestal. (tan 40° = 0.8391, √3 = 1.732)

11. The top of a 15 m high tower makes an angle of elevation 60° with the bottom of an electrical pole and angle of elevation of 30° with the top of the pole. What is the height of the pole?

12. From the top of a tower 50 m high, the angles of depression of the top and bottom of a tree are observed to be 30° and 45° respectively. Find the height of the tree. (√3 = 1.732)

13. A man is watching a boat speeding away from the top of a tower. The boat makes an angle of depression of 60° with the man's eye when at a distance of 200 m from the tower. After 10 seconds, the angle of depression becomes 45°. What is the approximate speed of the boat (in km/hr), assuming that it is sailing in still water? (√3 = 1.732)

14. An aeroplane at an altitude of 1800 m finds that two boats are sailing towards it in the same direction. The angle of depression of the boats as observed from the aeroplane are 60° and 30° respectively. Find the distance between the boats. (√3 = 1.732)

15. From the top of a lighthouse, the angles of depression of two ships on opposite sides of it are observed to be 30° and 60°. If the height of the light house is h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is ^4h/_√3 m.

16. From the top of a 12 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.

17. A pole 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point 'A' on the ground is 60° and the angle of elevation to the top of the tower is 45°. Find the height of the tower. (√3 = 1.732)

18. From the top of a tree of height 13 m the angle of elevation and depression of the top and bottom of another tree are 45° and 30° respectively. Find the height of the second tree. (√3 = 1.732)

19. The angles of elevation and depression of the top and bottom of a lamp post from the top of a 66 m high apartment are 60° and 30° respectively. Find:
(i) The height of the lamp post.
(ii) The difference between height of the lamp post and the apartment.
(iii) The distance between the lamp post and the apartment. (√3 = 1.732)

20. A man is standing on the deck of a ship, which is 40 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Calculate the distance of the hill from the ship and the height of the hill. (√3 = 1.732)